Square Peg problem
The square peg problem is deceptively simple: Given a continuous simple closed curve in the plane, prove that there are four points on the curve that form a perfect square. It was first posed in 1909, and remains open despite a long history of papers on the subject. The history of work on the subject reveals that it is relatively simple to prove that a differentiable closed curve has an inscribed square, but it is very difficult to extend the required arguments to the case where the curve is merely continuous. (For instance, does a closed fractal such as the Koch snowflake contain an inscribed square? In general, this is unknown.)
Results on Inscribed Squares
With Elizabeth Denne of Smith College and John McCleary of Vassar College, I have worked on the problem from a new angle. Our results prove that there are an odd number of squares in any simple closed curve which is differentiable or ``not too rough``. Here is an example of such a curve, with 5 inscribed squares.
Arnold categorized immersed plane curves in terms of three invariants called Strangeness St, positive Jump J + and negative Jump J − . In these terms, embedded loops like the curve above have all three invariants equal to zero. It is interesting to look at the squares inscribed in some example curves with different values of these invariants.
For instance, this figure 8 curve with St = 0,J + = 0,J − = − 1 has two inscribed squares.
This curve, created by performing a Reidemeister II move on a loop, has St = 0,J + = 0,J − = − 2 and seems to have 9 inscribed squares. I conjecture that only 3 of these are topologically nontrivial, while the other 6 cancel each other out pairwise. (I admit that the square in the upper right and the one in the lower left are pretty close to one another. But according to my computer program, they really are different.)
This curve has St = 1,J + = − 2,J − = − 4 and 14 squares
This curve has St = 2,J + = − 4,J − = − 6 and 15 squares
This curve has St = 3,J + = − 6,J − = − 8 and 12 squares
Based on these examples, jump and strangeness seem to play a role. We might guess that the number of squares is equal to St + (J + − J − ) + 1 mod 2. Unfortunately, our theory is only a mod 2 theory, so no further results can be drawn from it as yet. However, it is clear that there is more to see: the figure 8 curve above must have a square inside each lobe of the 8, so though the theory only predicts that the number should be even we know that the number must be at least 2.
Results on Inscribed Triangles
The Ellipsea on a plane curve, you can find b and c so that Δabc is an equilateral triangle. This is true for any shape of triangle (but you cannot guarantee a particular size for the triangle). We conjecture that these triangles can be chosen so that they vary continuously with a. That is, that there is a continuous loop of equilateral triangles inscribed in any plane curve. This animation (requires Quicktime Player) shows the loop of equilateral triangles in the ellipse. A higher resolution version of the movie is available as well.
We can visualize the curve of equilateral triangles in the ellipse as a curve in the three dimensional torus , where the three coordinates give the positions of the vertices along the ellipse. If we parametrize the torus as a cube with opposite faces identified, we can see the curve as the space curve show below:
A nonconvex curve
In the jellybean-shaped curve shown below, our computational experiments reveal that there are seven loops of equilateral triangles, one of which is topologically nontrivial (the basepoint travels all the way around the curve) and six of which are topologically trivial loops (the basepoint does not travel all the way around the curve).
The topologically trivial loops can be divided into two classes: in class A, the triangles remain near the bottom lobe of the curve. There are three loops in this class, all containing the same triangles, but with different basepoints. In class B, the triangles remain near the upper lobe of the curve. Again, there are three copies of the loop, each putting the basepoint in a different location.
Animations are available here for the trivial loops in class A, here for the trivial loops in class B, and here for the nontrivial loop. These animations are also available at higher resolution. The nontrivial loop is here, click here for class A, and here for class B.
It is also interesting to visualize the curve of equilateral triangles inscribed in the jellybean curve as a curve in the configuration space of triples of points on the jellybean. This space is a version of the 3-dimensional torus, which can be seen as a cube with top and bottom, left and right, and front and back faces identified. The pictures below show the seven loops of equilateral triangles in this cube.
In these images, the main loop is colored green, the trivial loops in class A are red, and the trivial loops in class B are blue. The second picture shows the threefold symmetry of the curve (seen by relabeling an equilateral triangle). We can see in the pictures that the main (green) loop represents the homology class (1,1,1) in the homology of the three-torus, which the other loops are homologically trivial in the three-torus.
We note that in these pictures, some of the trivial loops protrude from the cube. Because of the identifications which make the cube into the three-torus, these curves should appear as a collection of fragments which are glued together by the identifications on the sides of the cube. However, this makes the loops hard to see and also obscures the threefold symmetry of the picture. So we have lifted these loops into a covering space of the three-torus in order to draw them as connected space curves.
A figure-8 curveanimation (or look at this larger version). Somewhat surprisingly, this loop of triangles includes 'infinitesimal triangles', which happen when all three vertices approach the self-intersection of the curve. However, as we can see from the pictures below, this does not affect the homology class of the loop of triangles. The second picture, showing the threefold symmetry of the loop of triangles, is particularly interesting. It shows that while the curve may cross a subdiagonal of the cube (where two vertices come together on the curve at the intersection point), it never crosses the main diagonal (that is, we never have a triangle where all three vertices come together at the same parameter value.
As before, you can view these curves in 3d with these Geomview models.
A Reidemeister II curve
This curve is the result of a single Reidemeister II, or a pair of Reidemeister I moves, applied to a simple closed curve. We note that again, the loop represents the same homology class in the 3-torus, again the curve of equilateral triangles is 3-fold symmetric, and (as in the case of the jellybean curve) there are two classes of trivial loops on the curve.
As before, 3d models of these curves are available, too.